The Atlantic meridional overturning circulation (AMOC) is an important component of the climate system, of which change in the strength can affect meridional heat distribution between the northern and southern hemispheres. Proxy records show that changes in Atlantic Ocean circulation during the Late Pleistocene is associated with precessional cycle, but its physical mechanism remains unclear. Here we use a fully coupled climate model to investigate dynamics associated with AMOC changes in precessional band under glacial-interglacial climate conditions. Our results show that increase in boreal summer insolation can effectively weaken the AMOC during warm interglacial periods, while this weakening effect is reduced under glacial maximum. We further demonstrate that during the warm interglacial period increase in boreal summer insolation leads to sea surface warming and subpolar rainfall increase in North Atlantic, which jointly reduces sea surface density and hence the strength of deep water formation. During the glacial maximum period, climate responses to precessional change is of anti-phase impacts on the AMOC. At the low latitudes, a low pressure anomaly triggered by subtropical warming weakens atmospheric moisture export from the subtropical Atlantic to Pacific, increasing in net precipitation and hence freshening tropical sea surface in the North Atlantic. At the high latitudes, the warming-induced sea ice retreat promotes ocean heat loss via the enlarged ice-free area, and hence tends to strengthen the vertical mixing. The combined effects of low- and high-latitude responses finally leads to a trivial weakening of the AMOC. Overall, our results provide a systematic understanding of governing mechanism for precessionally-induced AMOC change under glacial-interglacial climatic backgrounds, shedding light on our interpretation of precessional periodicity in reconstructed ocean circulation changes during the Pleistocene.